Is there a name for this figure with alternating colors in the complementary set of Sierpinski's triangle?

Question

Start constructing Sierpinski triangle. On odd step - remove paint center white. On even step paint center blue. This figure has areas with arbitrary small patches of color, yet, area of both white and blue color is greater than zero. Lebesgue measures of white and blue colors are probably 4/7 and 3/7 respectively.

The resulting figure (first 6 iterations) is shown below.
What is the name of this object? Has it been studied?

Answer 1

Your question adresses the way to "access" (with its Computer Science meaning: the way to reach data) triangles in the complementary set of Sierpinski's triangle.

Here is an answer to your issue; I haven't gone into a full description, but I think that all the ideas are there.

It is based on two concepts:

• barycentric coordinates,

• their binary decompositions.

Let $ABC$ be the filled triangle in which the Sierpinski set is defined.

Let $$\begin{cases}u&=&u_0.u_1u_2u_3\cdots,\\ v&=&v_0.v_1v_2v_3 \cdots,\\ w&=&w_0.w_1w_2w_3\cdots\end{cases}$$

be the barycentric coordinates of a generic point $M$, expressed in bicimals (binary digits). Then

• (a) $u_0,v_0,w_0$ are equal to $1$ iff $M=A, B$ or $C$. In such cases, all the other $u_i,v_i,w_i$ are zero.

• (b) $M$ belongs to Sierpinski's triangle if and only if $u_i + v_i + w_i = 1$ for all $i$.

Property (a) is evident. Property (b) is presented in this Wikipedia article).

Let us consider for example a recursion level of $6$, the center of each elementary triangle can be described by the placement of zeros in the sequence:

$$R=\begin{pmatrix}u_1&u_2&u_3&u_4&u_5&u_6\\ v_1&v_2&v_3&v_4&v_5&v_6\\ w_1&w_2&w_3&w_4&w_5&w_6\end{pmatrix}$$

In the following graphical representation, an "approximation" of Sierpinski triangle, is represented by small red triangles.

With the example given in its legend, I think it will not be difficult to deduce a general description of the set of triangles you are interested in.

(see below the Matlab program which has generated the figure).

Fig. 1: The yellow triangle delimitated by the black lines is completely characterized by the following set of conditions on the barycentric coordinates of its points: $0<u<1/4, \ 0<v<1/4, \ 1/2<w<3/4$, which can be expressed on the binary decompositions of $u,v,w$ under the form $u_1=0, \ u_2=0, \ v_1=0, $$ \ v_2=0, \ w_1=1, \ w_2=0$.

I leave you the final task. i.e., defining which arrangements of bicimals give precisely the triangles you want to describe, being understood that placing a constraint on the bicimals situated for example on the right of array $R$ will have an incidence on the smallest triangles.

Matlab program for this figure:

clear all;close all;hold on;axis equal;
v=exp(i*2*pi/3*(0:2)); % cubic roots of 1
p=6;e=2^p;
for a=0:e;
    for b=0:e-a-1;
        c=e-a-b-1;
        % (a/e,b/e,c/e) = barycentric coordinates.
        R=[dec2base(a,2,p)-48;
           dec2base(b,2,p)-48;
           dec2base(c,2,p)-48];
        % Column R(:,k) contains "bicimals" uk,vk,wk
        t=sum(sum(R));
        if t==p;col='r';end; % (red) Sierpinski triangle
        if R(:,1)'==[0,0,0];col='g';end; % green tr.
        if sum(R(:,1))==1;col='y';end; % yellow tr.
        if sum(R(:,2))==1;col='b';end; % blue tr.
        z=a*v(1)+b*v(2)+c*v(3);
        fill(real(z+v),imag(z+v),col,'edgecolor','none');
    end;
end;